Higher-order derivatives

What is it?

Formulas

$$f''(x) = \frac{\dd^2 f}{\dd x^2}, \qquad f^{(n)} = \big(f^{(n-1)}\big)'$$

$$s(t) \xrightarrow{\ \dd/\dd t\ } v(t) \xrightarrow{\ \dd/\dd t\ } a(t) \xrightarrow{\ \dd/\dd t\ } j(t)$$

position, velocity, acceleration, jerk

Where it shows up in computing

• Kinematics: position, velocity, acceleration★★★★★fundamentalRobotics and control

  Acceleration is the second derivative of position; Newton's law $F = ma$ is about it.

• Trajectory optimization and MPC★★★★★frequentRobotics and control

  Minimum-jerk trajectories minimize $\int \dddot x^{\,2}\,\dd t$ for smooth robot motion.

• Bézier curves and splines★★★★★frequentComputer graphics

  Matching second derivatives at joints ($C^2$) removes visible kinks in fonts and CAD curves.

Where it shows up in AI

• Second-order (Hessian-based) optimization★★★★★fundamentalAI and machine learning

  Second derivatives measure curvature; Newton-type methods use them to choose the step.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• Critical points and derivative tests
• Taylor polynomial
• Hessian matrix
• Curvature (basic differential geometry)